1. For statements $p, q$ and $r$, use a truth table to show that each of the following pairs of statements is logically equivalent.
a. $(p \wedge q) \Leftrightarrow p$ and $p \Rightarrow q$.
b. $p \Rightarrow(q \vee r)$ and $\neg q \Rightarrow(\neg p \vee r)$.
c. $(p \vee q) \Rightarrow r$ and $(p \Rightarrow q) \wedge(q \Rightarrow r)$.
Answer (1)
Construct truth tables for each pair of statements.
Compare the truth values of the statements in each pair.
If the truth values are identical for all combinations, the statements are logically equivalent.
The pairs of statements in (a), (b), and (c) are logically equivalent. L o g i c a ll y Eq u i v a l e n t
Explanation
Understanding the Problem We are given three pairs of logical statements involving propositions p, q, and r. Our task is to demonstrate that each pair of statements is logically equivalent by constructing and comparing their truth tables.
Part (a): Setting up the Truth Table For part (a), we need to show that ( p ∧ q ) ⇔ p is logically equivalent to p ⇒ q . This means we need to construct a truth table that includes columns for p, q, p ∧ q , ( p ∧ q ) ⇔ p , and p ⇒ q .
Part (a): Constructing the Truth Table Here's the truth table for part (a):
p
q
p ∧ q
( p ∧ q ) ⇔ p
p ⇒ q
T
T
T
T
T
T
F
F
F
F
F
T
F
T
T
F
F
F
T
T
Part (a): Conclusion Observe that the truth values for ( p ∧ q ) ⇔ p and p ⇒ q are identical for all possible combinations of truth values for p and q. Therefore, the two statements are logically equivalent.
Part (b): Setting up the Truth Table For part (b), we need to show that p ⇒ ( q ∨ r ) is logically equivalent to ¬ q ⇒ ( ¬ p ∨ r ) . We will construct a truth table with columns for p, q, r, q ∨ r , p ⇒ ( q ∨ r ) , ¬ q , ¬ p , ¬ p ∨ r , and ¬ q ⇒ ( ¬ p ∨ r ) .
Part (b): Constructing the Truth Table Here's the truth table for part (b):
p
q
r
q ∨ r
p ⇒ ( q ∨ r )
¬ q
¬ p
¬ p ∨ r
¬ q ⇒ ( ¬ p ∨ r )
T
T
T
T
T
F
F
T
T
T
T
F
T
T
F
F
F
T
T
F
T
T
T
T
F
T
T
T
F
F
F
F
T
F
F
F
F
T
T
T
T
F
T
T
T
F
T
F
T
T
F
T
T
T
F
F
T
T
T
T
T
T
T
F
F
F
F
T
T
T
T
T
Part (b): Conclusion Observe that the truth values for p ⇒ ( q ∨ r ) and ¬ q ⇒ ( ¬ p ∨ r ) are identical for all possible combinations of truth values for p, q, and r. Therefore, the two statements are logically equivalent.
Part (c): Setting up the Truth Table For part (c), we need to show that ( p ∨ q ) ⇒ r is logically equivalent to ( p ⇒ r ) ∧ ( q ⇒ r ) . We will construct a truth table with columns for p, q, r, p ∨ q , ( p ∨ q ) ⇒ r , p ⇒ r , q ⇒ r , and ( p ⇒ r ) ∧ ( q ⇒ r ) .
Part (c): Constructing the Truth Table Here's the truth table for part (c):
p
q
r
p ∨ q
( p ∨ q ) ⇒ r
p ⇒ r
q ⇒ r
( p ⇒ r ) ∧ ( q ⇒ r )
T
T
T
T
T
T
T
T
T
T
F
T
F
F
F
F
T
F
T
T
T
T
T
T
T
F
F
T
F
F
T
F
F
T
T
T
T
T
T
T
F
T
F
T
F
T
F
F
F
F
T
F
T
T
T
T
F
F
F
F
T
T
T
T
Part (c): Conclusion Observe that the truth values for ( p ∨ q ) ⇒ r and ( p ⇒ r ) ∧ ( q ⇒ r ) are identical for all possible combinations of truth values for p, q, and r. Therefore, the two statements are logically equivalent.
Final Answer In summary, by constructing truth tables for each pair of statements, we have demonstrated that the statements in each pair are logically equivalent because their truth values match for all possible combinations of truth values of the individual propositions.
Examples
Understanding logical equivalences is crucial in computer science for simplifying complex Boolean expressions in program design. For instance, consider a program that checks if a user is eligible for a discount based on age and membership status. Instead of using a complex conditional statement, we can simplify it using logical equivalences to make the code more readable and efficient. This ensures the program behaves as expected while reducing the chances of errors.
